r*sin(b)/cos(a+b), B

Average Error: 13.9 → 0.4

Time: 6.7s

Precision: 64

Math

\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]

↓

\[r \cdot \left(\frac{\sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\right)\]

C

double code(double r, double a, double b) {
	return (r * (sin(b) / cos((a + b))));
}

↓

double code(double r, double a, double b) {
	return (r * ((sin(b) / (((cos(a) * cos(b)) * (cos(a) * cos(b))) - ((sin(a) * sin(b)) * (sin(a) * sin(b))))) * ((cos(a) * cos(b)) + (sin(a) * sin(b)))));
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 13.9

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
2. Using strategy rm

3. Applied cos-sum 0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
4. Using strategy rm

5. Applied flip-- 0.4

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)}{\cos a \cdot \cos b + \sin a \cdot \sin b}}}\]

6. Applied associate-/r/ 0.4

   \[\leadsto r \cdot \color{blue}{\left(\frac{\sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\right)}\]

7. Final simplification 0.4

   \[\leadsto r \cdot \left(\frac{\sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)} \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)\right)\]

Reproduce

herbie shell --seed 2020066 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))